We study the cycle structure of words in several random permutations. We assume that the permutations are independent and that their distribution is conjugation invariant, with a good control on their short cycles. If, after successive cyclic simplifications, the word $w$ still contains at least two different letters, then we get a universal limiting joint law for short cycles for the word in these permutations. These results can be seen as an extension of our previous work (Kammoun and Maïda. Electron. Commun. Probab. 2020;25:1-14.) from the product of permutations to any non-trivial word in the permutations and also as an extension of the results of Nica (Random Struct. Algorithms1994;5:703-730.) from uniform permutations to general conjugation invariant random permutations. In particular, we get optimal assumptions in the case of the commutator of two such random permutations.

中文翻译：

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共轭不变随机排列中单词的小循环结构

我们研究几种随机排列中单词的循环结构。我们假设排列是独立的，并且它们的分布是共轭不变的，并且可以很好地控制它们的短周期。如果在连续的循环简化之后，单词$w$仍然包含至少两个不同的字母，那么我们就得到了这些排列中单词的短周期的通用限制联合律。这些结果可以看作是我们之前工作（Kammoun and Maïda. Electron. Commun. Probab. 2020;25:1-14.）的延伸，从排列的乘积到排列中任何不平凡的单词，也可以看作是Nica ( Random Struct. Algorithms 1994;5:703-730.)结果从均匀排列扩展到一般共轭不变随机排列。特别是，在两个这样的随机排列的换向器的情况下，我们得到了最优假设。